You can absolutely do the above said task with the shield that you provided information about.
All you have to do is to connect the Shield to the Arduino and upload the WiFiWebServer sketch from the Arduino's examples in the WiFi library!
Then you'll have to replace the SSID and Password with your WiFi's password.
Once uploaded, then you'll get your Arduino's (Server's) IP Address on the Serial monitorl.
Note the IP Address and open any browser on any device of your choice.
Your device must be connected to the same WiFi network to which your Arduino is connected to. If you wish to access the Arduino server from outside your LAN then you'll have to forward/open a port on your WiFi router and direct it to the IP Address of your Arduino. But I am assuming that you just want to access it from your LAN therefore the port forwarding can be ignored.
You'll see the default webpage in the Sketch which you can change/edit as per your own wish by using HTML.
Here's the link to the detailed explanation from arduino.cc itself.
Coming to the Second part of your question: "Also, once a device (say a phone) is connected, how would it go about requesting a web page, or other information?"
Well once the server is started on the Arduino all you do is type the IP of Arduino in the browser of the device and then the browser sends a HTTP request to the server.(This is the same way in which the browser sends requests to any other website or server like google/youtube/stackexchange . It is same for all ).